Finite axiomatizability of quasi-normal modal logics

Abstract

Quasi-normal modal logics are logics in a modal language that contain the logic ${\bf K}$, are closed according to the modus ponens rule, and for which is not postulated Godel's rule. Until recently, little attention was paid to these logics, despite the fact that among the first systems of modal logics formulated by C.I. Lewis, there were also quasi-normal logics. In this paper, we consider the question of finite axiomatizability of quasi-normal modal logics.

As is well known, the quasi-normal partner of the logic ${\bf K}$ does not have a finite axiomatization. In addition, there are other modal normal finitely axiomatizable logics, whose quasi-normal partners have no finite axiomatization. (An example of such logic is the logic ${\bf D}$.) Therefore, the question of the finite axiomatizability of a particular modal quasi-normal logic is not trivial.

Note that the well-known special criteria for the finite axiomatizability of quasi-normal logics concern only quasi-normal partners of normal modal logics.

In this paper, a generalization of these particular criteria is obtained for the case of arbitrary quasi-normal modal logics. Thus, we obtain a special criterion of finite axiomatisability applicable both for quasi-normal partners of normal logics and for quasi-normal logics which are not a quasi-normal partner of any normal logic.

In addition, a method for constructing a possible finite axiomatization of these quasi-normal finitely axiomatizable logics is given. We also present an algorithm that gives an absolute axiomatization of the logic $L$ according to the available relative axiomatization of the quasi-normal logic $L$ over the quasi-normal partner of the logic ${\bf K}$.

Separately, the axiomatization of extensions of the logic ${\bf K4}$ is considered. A special criterion for the finite axiomatizability of extensions of this logic is formulated. We present an algorithm that gives an absolute axiomatization of the logic $L$ by the available relative axiomatization of the quasi-normal logic $L$ over the quasi-normal partner of the logic ${\bf K4}$.